1 Introduction

During an earthquake a large amount of energy through ground motion is imparted to structures. In the basic concept of conventional inelastic design, structural members should absorb or dissipate these transmitted ground motions by inelastic cycling deformation however, the structures are not supposed to suffer a major collapse. In order to reduce inelastic energy dissipation demand on the framing system of structures, passive energy dissipation devices are used in new structures as well as the rehabilitation of ageing or deficient structures. The use of energy dissipation systems reduces damage to the frame systems. A number of passive energy dissipating systems such as the yielding of metals, frictional sliding and phase transformation in metals, deformation of visco-elastic fluids and solids, and fluid orificing are commercially available or under development (Symans et al. 2008).

A fluid viscous damper, which is one of the passive damper devices, consists of cylindrical piston immersed in a viscous fluid. This type of damper, widely used in aerospace and military applications, has recently been adapted for building applications as shown in Fig. 1 (Constantinou and Symans 1992). The characteristics of these devices are the linear viscous response achieved over a broad frequency range, insensitivity to temperature and compactness in comparison to the stroke and output force. This type of damper dissipates energy through the movement of the piston in the highly viscous fluid. If the fluid is purely viscous, then the output force of the damper is directly proportional to the velocity of the piston (Ashour and Hanson 1987; Hahn and Sathiavageeswara 1992).

Fig. 1
figure 1

Construction of fluid viscous damper (Constantinou and Symans 1992)

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An optimization technique must be employed to determine the location of dampers and distribution of damper factors. In the early stages of the development of optimization techniques, the configuration and location of dampers were investigated on the basis of the minimization of an energy criteria (Gürgöze and Müller 1992), minimization of a controllability index (Zhang and Soong 1992; Shukla and Datta 1999), minimization of the sum of storey stiffness (Tsuji and Nakamura 1996), a finite perturbation method (Cao and Mlejnek 1995), and using a simplified sequential search algorithm (Lopez and Soong 2002). Some versatile investigations have been developed for optimal damper configuration and location assisted by the concepts of inverse problem redesign approaches and optimality criteria based design approaches which require sensitivity formulations (Takewaki 1997, 1999, 2000, 2009; Takewaki and Yoshitomi 1998; Agrawal and Yang 2000; Singh and Moreschi 2001; Park et al. 2004; Wang et al. 2010; Levy and Lavan 2006; Lavan and Levy 2006; Cimellaro 2007; Lavan et al. 2008; Lavan and Levy 2009, 2010; Aydin et al. 2007; Aydin 2012; Fujita et al. 2010a, b). Gradient based methods need sensitivity information and these sensitivities may lead to numerical difficulties for various objective functions and constraints. This difficulty may be overcome using direct search optimization techniques because these techniques do not have mathematical requirements such as derivatives. Many research studies (Bishop and Striz 2004; Wongprasert and Symans 2004; Dargush and Sant 2005; Silvestri and Trombetti 2007; Lavan and Dargush 2009; Farhat et al. 2009) used different versions of genetic algorithm (GA) which is possibly the most popular direct search algorithm in optimal damper problems. The major disadvantages of GA are; that it needs much more functional evaluations in comparison to linearization methods; there is no guarantee of convergence even for local minima; and it is very slow and computationally expensive in models with many parameters.

One of the most recently developed direct search methods is the artificial bee colony algorithm (ABCA) which is a member of the family of swarm intelligence based algorithms. ABCA was first introduced by Karaboga (2005) to solve continuous optimization problems. Sonmez (2011a, b) used the modified version of the ABC algorithm to find solutions to continuous and discrete structural problems. He demonstrated that the ABCA provides results that are as good as or better than other optimization algorithms such as ant colony, genetic algorithms and particle swarm optimization. In addition, Sonmez stated that the ABCA shows a remarkably robust performance with a 100 % success rate. Although the ABCA does not show any significant improvement in the speed of convergence in terms of the number of function evaluations performed to obtain the best designs when compared to genetic algorithms, particle swarm optimization and harmony search, the ABCA has three main advantages: it is easy to implement, requires less control parameters and is robust.

The aim of this study was to develop a design optimization method to find the optimal size and location of dampers using the ABCA in planar building structures. The Fourier Transform was applied to the equation of motion then the transfer functions were derived to determine the response of the structure evaluated at the fundamental natural frequency of the structure. The objective functions were defined as the transfer function amplitude of the top storey displacement and the elastic base shear force at the fundamental natural frequency of the structure (Takewaki 2000; Aydin et al. 2007). In other words, the first mode responses based on transfer functions were minimized under design constraints. Design variables were chosen as the damping coefficients of the added dampers and there was an active constraint on the total added damping coefficients. Furthermore, it was possible to select the passive constraints on the upper and lower bounds of the damping coefficient of the added dampers. The proposed ABCA for finding the optimal damper placement was compared with the steepest direction search method. The verified design method with the gradient procedure was applied to rehabilitate an irregular nine floor building with soft storeys. The results of the optimal damper methods were tested using time history analyses of the Kobe (NS) earthquake ground motion record.

2 Definition of structural response based on transfer functions

The derivation of transfer functions is presented to define the structural response. The transfer functions used in this study were those of Takewaki (2000). An N-storey frame planar building as shown in Fig. 2 was the model for the design of the placement of the added dampers. The Fourier Transform was applied to the equation of motion and the structural response was defined at the fundamental natural frequency of the structure in the studies mentioned above. The transfer function vector of the displacements evaluated from the fundamental natural frequency of the structure was presented by Takewaki (2000) as follows:

$$ \widehat{\boldsymbol{U}}\left( {\omega_1 } \right)=-\boldsymbol{A}^{-1}\boldsymbol{Mr} $$
(1)

where M denotes the mass matrix, r is the influence vector in terms of the direction of the base acceleration. The matrix A is given as

$$ \boldsymbol{A}=\boldsymbol{K}+i\omega_1 \left( {\boldsymbol{C}+\boldsymbol{C}_{\boldsymbol{ad}}} \right)-\omega_1^2 \boldsymbol{M} $$
(2)

where \(\boldsymbol {K}\), \(\boldsymbol {C}\), \(\boldsymbol {C_{ad}}\) and \(\omega _{1}\) represent the global stiffness matrix, structural damping matrix, added damping matrix and the first natural circular frequency of the structure, and i denotes \(\sqrt {-1} \). If the transfer function vector of the displacements, given in (1), is multiplied by the stiffness matrix, the transfer function vector of elastic forces can be defined (Aydin et al. 2007) as:

$$ \boldsymbol{F}\left( {\omega_1 } \right)=-\boldsymbol{KA}^{-1}\boldsymbol{Mr} $$
(3)
Fig. 2
figure 2

N-Storey moment frame planar steel building with local coordinate systems

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3 Description of the optimal damper problem

In the damper optimization methods, some of the objective functions are top displacements, maximum inter-storey drifts, the sum of interstorey drifts (Takewaki 2000), base shear (Aydin et al. 2007), a defined damage index based on energy (Lavan and Levy 2005), absolute acceleration (Cimellaro 2007), overturning moments, a defined damage index, and combinations of some structural performance functions. Various objective functions can be used in order to solve an optimal damper problem and the importance of various cost functions can increase for different types of structures. While a decrease in displacements or inter-storey drifts is important for a displacement-based design, some internal forces and accelerations can be important for a forced-based design. In other words, a defined structural damage index or an energy index may be important for different structures. In this respect, a slight variation of the objective function to solve an optimal damper problem also changes the results of the damper design.

In this study, the optimal damper problems were based on minimizing the top displacement and the elastic base shear response of the structure. The mathematical statement of the damper design optimization in the context of this paper is as follows:

$$ \textrm{Minimize}\qquad f({\boldsymbol{C_{ad}}})=f({c_1 ,c_2 ,c_3,\ldots ,c_N}) $$
(4)

Taking into account the inequality constraints on the upper and lower bounds of the damping coefficients of each added damper gives the following:

$$ 0\le c_i \le \bar c_i \quad \left( {\mathrm{i}=1,2,\ldots ,\mathrm{N}} \right) $$
(5)

where \(\bar c_i \) is the upper bound of the damping coefficient of the damper in the ith storey and an equality constraint on the sum of the added dampers is written as:

$$ \sum_{i=1}^N {c_i =\bar C } $$
(6)

where \(\bar C \) is the sum of the damping coefficients of the added dampers.

3.1 Minimization of top displacement

The first optimization problem based upon the minimization of the norm of the top displacement transfer function is expressed as follows:

$$ f_1 \left( {\boldsymbol{C_{ad}} } \right)=\left| {\widehat{U}_t \left( {c_i } \right)} \right|\quad \left( {\mathrm{i}=1,2,...,\;\mathrm{N}} \right)\;\textrm{Minimize} $$
(7)

where \(\left | {\widehat {U}_t \left ( {c_i } \right )} \right |\) corresponds to the transfer function amplitude of the top displacement evaluated at the undamped fundamental natural frequency, which is in vector \(\widehat {\boldsymbol {U}}\left ( {\omega _1 } \right )\). This objective function is equal to the sum of the transfer function amplitudes of the inter-storey drifts evaluated at the undamped fundamental natural frequency (Takewaki 2000).

3.2 Minimization of elastic base shear

The second optimization problem based on the minimization of the norm of the elastic base shear transfer function is expressed as follows:

$$ \textrm{Minimize}\; f_{2} \left( {\boldsymbol{C_{ad}} } \right)=\left| {V_b (c_i )} \right|\quad \left( {\mathrm{i}=1,\;2\,,...,\;\mathrm{N}} \right) $$
(8)

where,

$$ \left| {V_b (c_i )} \right|=\left| {F_1 (c_i )} \right|+\left| {F_2 (c_i )} \right|+\ldots +\left| {F_N (c_i )} \right| $$
(9)

where \(\left | {F_i } \right |\) is the transfer function amplitude of the ith storey elastic force evaluated at the undamped fundamental natural frequency, which is in vector \(\boldsymbol {F}({\omega _1})\). This performance function was proposed by Aydin et al. (2007).

4 The artificial bee colony algorithm (ABCA)

4.1 Bee behaviour in nature

Nature inspired algorithms based on the social behaviour of certain insects can solve many complex optimization problems. These insects live together in their nest and divide up work tasks such as foraging, nest building and guarding the colony. Each member of the colony performs their tasks by interacting or communicating in a direct or indirect manner in their local environment (Bonabeau et al. 1999; Kennedy et al. 2001). Like other social insects, honeybees are only able to survive as a member of a community known as a colony. A bee colony contains three types of bees; a queen bee, which is a fertile female, thousands of sterile female worker bees and a few thousand drones, which are fertile males. The task of queen is to mate with the drones, lay eggs and start new colonies. All the other tasks associated with colony life such as; building the honeycomb and storing food, cleaning cells, feeding the queen and drones, and guarding the entrance are performed by the female workers. When the female workers are about three weeks old, they cease performing most tasks within the hive and become foragers for the rest of their lives. The forager honeybees search for promising flower nectar from different food sources in multiple directions up to 12 kilometres from the hive, but they mostly fly within a three-kilometre radius (Honey Bee Biology 2010). Von Frisch (1967) discovered that when honeybees find a food source they advertise their findings by performing a “round dance” or a “waggle dance” based on the distance of the food source from the hive. The round dance gives the information that the discovered food source is in the vicinity of the hive and the waggle dance informs the other bees that the food source is situated more than 100 m from the hive. The direction and duration of dances are closely correlated to the direction and distance of the patch of flowers. The longer the duration of the dance, the better the food source quality. Hence, a bee indicating a high quality food source has a greater probability of being selected by other bees as it dances for longer (Karaboga and Basturk 2008). Figure 3 depicts one hive, three food sources and the waggle dance performed by the bees that discovered these food sources (Lemmens et al. 2007). All the forager bees will find food sources of different quantities during their trip. Hence, after unloading the nectar, each bee can follow one of three options: (a) abandon the food source and search for another promising flower patch, (b) continue to forage at the food source without recruiting their nest mates, or (c) perform the one of the bee dances to recruit nest mates before returning to the food source. The option selected is based on the quantity of food of the nectar source found by the individual bee. If a bee finds a nectar source that is above a certain limit, she follows option (c). If the nectar source is average, the bee goes to forage at the food source without recruiting nest mates (option b). Otherwise, the bee continues to search for promising nectar sources (option a). The main goal of the bees is to locate the most abundant nectar source (Karaboga and Basturk 2008).

Fig. 3
figure 3

Waggle dances for different food sources (Honey Bee Biology 2010)

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4.2 ABCA for damper design

In the ABCA, the food sources and their nectar quantity represent possible solutions to a given optimization problem. The location of the food source corresponds to the design variables \(\boldsymbol{C}_{ad} (c_{1}, c_{2}, \ldots c_{N} )\). In the first step of the ABCA, all the forager bees (BN) leave the hive to search for promising flower patches. The location of a food source, s, can be determined as:

$$ \textrm{~}_{s}c_j^{{new}} =c_j^{{low}} +\gamma \left( {c_j^{\textit{up}} -c_j^{{low}} } \right) $$
(10)
$$ c_j^{{low}} \le c_j \le c_j^{\textit{up}} \quad \quad \quad \quad j=1,2,\ldots ,N $$
(11)

where \(\gamma \) is a random number between 0 and 1. \(c_j^{{low}} \) and \(c_j^{\textit {up}}\) are the lower and upper bounds of the jth design variable as in (11), respectively.

After the forager bees return to the hive with a certain amount of nectar (\(f_{h})\), determined using (7) or (8), the first half of the bees (SN) which found the best food sources become “employed bees.” The remainder of the bees (called unemployed or onlooker bees) watch the dancing bees to decide one of which employed bee they will follow. The unemployed bees select a food source according to a probability proportional to the amount of nectar to be found at that food source (Von Frisch 1967). The probability \(P_{s}\) for source s is calculated in the following way:

$$ P_s =\frac{1/\textrm{~}_{s}f_{h} (\boldsymbol{C_{ad}} )}{\sum_{j=1}^{SN} {1/\textrm{~}_{s}f_{h}(\boldsymbol{C_{ad}} )} } $$
(12)

Each food source has only one employed bee; that is, the number of food sources is equal to the number of employed bees. The number of unemployed bees that will fly to a food source depends upon the amount of nectar at the source. The unemployed bees choose a food source according to the quantity of the nectar. More unemployed bees will choose to visit an abundant nectar source while fewer or no unemployed bees will choose the food source having less nectar.

After an employed bee has recruited unemployed bees, if any, she leaves the hive to find a better food source (called candidate food sources) in the neighbourhood of the previous food sources discovered by her and other employed bees. This means that the ABC algorithm uses the previous food source \(\left ( {\textrm {~}_{s}c^{old}} \right )\) to search for a candidate food source \(\left ( {\textrm {~}_{s}c^{new}} \right )\). Numerically, the location of a candidate food source, s, is determined as:

$$ \textrm{~}_{s}c_j^{{new}} =\textrm{~}_{s}c_j^{old} +\phi \left( {\textrm{~}_{s}c_j^{old} -_{k}c_j^{old} } \right) $$
(13)

where \(\phi \) is a random number between \(-\)1 and 1. \(\textrm {~}_{s}c_j^{new}\) is an updated design variable. The left hand subscripts represent the solution number (food source, \(s=1,2,\ldots ,SN)\) while the right hand script denotes the design variable number \(\left ( {k=1,2,\ldots ,SN} \right )\). k is a randomly chosen integer number but cannot be equal to \(s. \,\,\textrm {~}_{s}c_j^{old} \) that plays an important role in the ABC convergence behaviour since it is used to control the exploration abilities of the bees. It directly influences the location of the new food source, which is based on the previous location of other food sources. If the quantity of food in the new location is better than old one, the new position becomes the food source; otherwise, the old location is maintained as the best food source.

As mentioned above, the ABC algorithm is iterative. If there is no improvement in the amount of nectar from a food source after a predefined iteration (LIMIT), this food source is discarded by its employed bee and this bee becomes a ‘scout bee’, one of the colony’s explorers. Primarily concerned with finding any kind of nectar source and not having any guidance as to where to look for food these scouts may accidentally discover rich, entirely unknown food sources and then they become an employed bee. A new location found by a scout bee is calculated using (10).

In general the ABC algorithm uses three control parameters i) the number of worker bees (BN) which leave the hive to search for promising flower patches, ii) a predefined iteration number (LIMIT) if there is no improvement in the amount of nectar from a food source after this number and iii) the maximum iteration number for searching for food (MCN). Unlike a genetic algorithm, the ABC algorithm does not require external parameters such as crossover and mutation rate.

4.3 Proposed ABCA for optimal damper design

The procedure for the placement of added dampers in a planar building frame is given as follows;

  1. Step 1.

    Set the total number of Bees (BN), maximum number of cycles (MNC), the predetermined limit value (LIMIT) and number of design variables (D). Generally, BN may be set to \(10\times D\). MNC is determined after a few runs of the problem. The predetermined limit value is set to zero for optimal damper problems.

  2. Step 2.

    Generate a random initial bee colony. Each design variable corresponding to the location of a bee in the solution space is generated between the permissible lowest and highest damper value, which is constructed using following equation:

    $$\begin{array}{lll} &&\textrm{~}_{s}c_j^{{new}} =c_j^{{low}} +\gamma \left( {c_j^{\textit{up}} -c_j^{{low}} } \right)\\ &&s=1,2,3,..SN \quad j=1,2,..,N \end{array}$$

    where \(c_j^{{low}}\) is set to zero. The total amount of added damper coefficient \(\left ( {\sum _{j=1}^n {c_j } } \right )\) must be equal to \(\bar C \) for each bee. If it is not equal, the c\(_{\textrm j}\)’s must be rearranged based on the following expression:

    $$c_j =\frac{\sum_{j=1}^n {c_j } }{\bar C }\ast c_j $$
  3. Step 3.

    Read the input data to construct the structure then calculate the global stiffness matrix (K), and the structural damping matrix (C). Then calculate the added damping matrix (C \(_{\textbf ad})\) and the first natural circular frequency of the structure (\(\omega \) \(_{1})\) for all bees (BN). The transfer function amplitude of the structural response (\(f_{h})\) may be calculated based on the selected objective function. For example; when it is required that the top displacement is minimized, the transfer function vector of the displacements evaluated from the fundamental natural frequency of the structure \(\widehat {\boldsymbol {U}}({\omega _1})\) is calculated. Then the corresponding top displacement \(\widehat {\boldsymbol {U}}_t ({\omega _1})\), which holds the objective function value, is selected from the displacement transfer vector. The sort bees base in the objective function value is in ascending form. This means that the lowest objective function value is placed in the first row of colony because the lowest value represents the minimum displacement.

  4. Step 4.

    Select the best half of the sorted bees (SN) from the candidate food sources (BN). The bees associated with the best locations then become employed bees. The remainder of the sorted bee become unemployed bees.

  5. Step 5.

    Set cycle\(=\)1;

  6. Step 6.

    Loop over each food source (\(s = 1,2,{\ldots },SN\))

    1. 6.1

      Determine the number of unemployed bees which visit the food location i based on the probability \(P_{s}\) using (12).

    2. 6.2

      Produce new solutions for an employed bee and any existing unemployed bees, using (13).

    3. 6.3

      Construct \(\boldsymbol {C_{ad}}\) matrices using new solutions then calculate the transfer function amplitude of the structural response (\(f_{h})\) for the employed and unemployed bees.

    4. 6.4

      Select the best food level, which means that minimum displacement value around food source i. If the quantity of food in the one of the new location is greater than the old one, then the new position becomes the food source; otherwise, old location is maintained as a food source.

  7. Step 7.

    cycle \(=\) cycle + 1

  8. Step 8.

    If the cycle is less than MNC or there is no improvement in the quantity of the food source in a certain cycle, go to step 6. Otherwise, stop the procedure.

It is noteworthy that the proposed ABCA given above has some deviations from the original ABC algorithm proposed by Karaboga and Basturk (2008). The first deviation is that the original algorithm selects the half of the bees as “employed bees” and generates solutions for these employed bees. On the other hand, the ABCA generates BN solutions initially and chooses best half of bees as the initial solutions. Bees visiting the best food locations will be “employed bees” for these food locations and the rest of the bees become “onlookers”. The second deviation is that in the original algorithm every employed bee constructs solutions in their neighbourhood and moves these locations if they have better fitness, then unemployed bees construct solutions in the neighbourhoods of the employed bees they followed. On the other hand, in ABCA employed bees and any existing onlooker bees visit the food locations found by the employed bees together. One of bees finding the best fitness in the neighbourhood will be an employed bee in the next cycle. By doing so, the greedy selection process is performed once in each cycle. The other deviation is that the employed bee associated with the best solution is never a scout bee even if even if there is no improvement in the nectar after the LIMIT number of cycles.

5 Comparison of the SDSA and ABCA methods

The paper presents the investigation of a new optimization technique (ABCA) for the optimal placement of added dampers in planar building frames. This section describes the verification with the gradient based method (SDSA) of the proposed ABCA. A nine-storey steel planar building model was chosen for the comparison. The objective functions defined in Sections 3.1 and 3.2 were minimized to determine the optimal damper design using both methods.

The steepest direction search algorithm used for the verification may be similar to the steepest descent method. However, while the conventional steepest descent method uses the gradient vector of objective function itself as the direction and does not use any optimality criteria. The steepest direction search algorithm takes full advantage of the derived optimality criteria, and does not adopt the gradient vector as the direction. It guarantees the satisfaction of the optimality criteria (Takewaki 1999). It is a simple and effective method to find the optimal damper placement however, this optimality criteria method needs the first and second order sensitivity formulations of the objective functions. When the base shear transfer function was selected as an objective function in the tall building, some convergence problems have been reported (Cimellaro 2007) for gradient-based solutions. The reason for this behaviour is that the gradient of objective function for base shear is generally affected by a change in scale of damping coefficient that can generate convergence problems. In the case when there are changes in the objective functions and constraints, the derivation of new sensitivity formulations is required and some convergence problems can be exposed because of the complex objective functions and constraints. The proposed artificial bee colony algorithm is an iterative method and it does not require any gradient formulation and its derivation. Moreover, when the final values of both objective functions are compared with the last value of the objective function of the gradient based method, the proposed algorithm gives a good performance. When seen from this perspective, the proposed method is a simple tool for finding optimal damper design.

5.1 Example 1

A nine-storey steel planar building was selected as the structural model as shown in Fig. 4. Each storey was 4 m high and the span of each bay was 8 m. Young’s Modulus was \(2.06 \times 10^{5}\) MPa in the columns and beams. The shear deformation of the structural elements was not taken into account. The properties of the structural elements are presented in Table 1. The undamped fundamental natural frequency of the model structure was computed as 5.239 rad/s. The critical damping ratio of the steel structure model was taken to be \(\xi =0.02\) in the undamped first natural frequency. The total damper capacity \(({\bar C})\) was considered to be \(2.646\times 10^{7}\) Ns/m and the added damping for each step is given as \(\Delta C=\bar C /m\) in the gradient based damper design. The number of design steps was 2000. The constraints on the upper bounds of the damping coefficients of the added dampers were assumed to be inactive.

Fig. 4
figure 4

Nine-storey steel planar building model

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Table 1 The properties of the structural members
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Firstly, the optimal damper placements for both the minimization of the top displacement and the base shear were determined using SDSA. Different distributions of the dampers were determined for the two performance functions as shown in Fig. 5a and b. The total damper capacity was distributed over the 2nd, 3rd and 4th storeys in the top displacement minimization as shown in Fig. 5a and it was spread over the 1st, 2nd, 3rd, 4th, 5th, 6th, 7th and 8th storeys in the base shear minimization as shown in Fig. 5b. The optimal design histories of the dampers for the two objective functions in terms of the transfer function amplitudes of the top displacement and the base shear are presented in Fig. 6a and b. Each optimal damper design minimizes its objective functions. The first order sensitivities of both objective functions are plotted in Fig. 7a and b. The convergences of the optimal design process were satisfied in all storeys for the top displacement optimization and the base shear optimization. When the dampers were concentrated on the storeys where the interstorey drifts were large, the amplitude of the top displacement was effectively reduced. When the dampers were placed to minimize the base shear, the damping coefficient of the added dampers was progressively reduced from the upper to the lower storeys, thus drastically reducing the base shear response.

Fig. 5
figure 5

Variation of the damping coefficient of the dampers for; a objective function as top displacement (Takewaki 2000) b objective function as base shear (Aydin et al. 2007)

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Fig. 6
figure 6

Variation of the transfer function amplitude for; a top displacement minimization b the base shear force minimization

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Fig. 7
figure 7

Variation of the first order derivatives with respect to design variables (a) Top displacement minimization (Takewaki 2000) (b) base shear minimization (Aydin et al. 2007)

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In order to make the comparison, the proposed ABCA to determine the optimal damper placement without using any gradient information was applied to a nine storey planar building frame as shown in Fig. 4. The design histories of the damper for the two objectives are shown in Fig. 8a and b for the ABCA. Due to the random numbers, more than one run of the algorithm should be performed to assess the random number effects therefore, ten independent runs were performed and the minimum (best), the maximum (worst) and the average of the objective functions are shown in Fig. 8a and b. It is significant that when the top displacement was used as an objective function, the convergence was observed after approximately 50 cycles; on the other hand, when the base shear was the objective function, at least 100 cycles were required before convergence occurred. Hence, the maximum number of cycles (MNC) was set to 150 for all cases. Furthermore, for all cases, the number of foraging bees in a colony was assumed to be 10 times the design variables.

Fig. 8
figure 8

Design history of optimal damper problem using ABCA (a) Top displacement minimization (b) Base shear minimization

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Table 2 presents the optimal damper designs for the SSDA and ABCA showing that the dampers for SDSA were distributed on the 2nd, 3rd and 4th storeys in the top displacement minimization and the dampers for the ABCA were placed predominantly on the 2nd, 3rd and 4th storeys in the top displacement minimization. The ABCA involves the selection of random numbers; therefore, for each optimization problem the operation was performed ten times and the best and the worst results were selected as shown in Table 2. In order to develop a more practical solution, the insignificantly damping coefficient of the dampers placed on the 1st, 5th, 6th, 7th, 8th and 9th storeys in the top displacement minimization and on the 9th in the base shear minimization can be considered to be zero. Table 2 shows an acceptable compatibility of the results of the two optimization methods. Furthermore, it can be seen that the proposed ABCA method, which does not require any gradient information, is effective in finding the optimal size and location of the added dampers.

Table 2 Comparison of the results of the optimal damper designs based on the ABCA and SDSA methods
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The penultimate and last lines of Table 2 show respectively; the transfer function amplitude of the top displacement (f\(_{1})\) and the transfer function amplitude of the base shear (f\(_{2})\) evaluated from the fundamental natural frequency of the structure. The results of the objective functions based on both top displacement minimization and base shear minimization show that both the best and the worst damper designs determined using the ABCA considerably decrease the structural response.

5.2 Example 2

The proposed ABCA method to find the optimal damper allocation was verified by a gradient base method (SDSA). The proposed damper design methods were carried out on three irregular planar building models as shown in Fig. 9a–c. The model structure used for the verification in example 1 was modified by adding diagonal bars. The cross section area of each diagonal brace was assumed to be 0.23607 m\(^{2}\). Diagonal bars were fixed to the first three storeys in accordance with the 123 bracing model as shown in Fig. 9a, to the 4th, 5th and 6th storeys in the 456 bracing model as shown in Fig. 9b, to the 7th, 8th and 9th storeys in the 789 bracing model as shown in Fig. 9c. The objective of the application of the added diagonal bars was to satisfy a stiffness irregularity in the planar building model. Then, to rehabilitate these irregular models with dampers the optimal damper placement was estimated with the ABCA. The effects of the variations in the position of the stiff storeys were investigated in the optimal damper design. Figure 10 shows the variation of the objective functions with respect to the number of cycles in all bracing models. When the ABCA was performed to optimally place the added dampers, random numbers were used, for this reason; the ABC algorithm was run at least ten times. As shown in Fig. 10, the worst, the best and the average performance values obtained from the ABCA were chosen to demonstrate that the convergence criteria were satisfied.

Fig. 9
figure 9

Nine-storey steel planar building frame with a 123 bracing model b 456 bracing model c 789 bracing model

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Fig. 10
figure 10

Design history of optimal damper problem in case of ABCA for; a Top displacement optimization with 123 bracing model b Top displacement optimization with 456 bracing model c Top displacement optimization with 789 bracing model d Base shear optimization with 123 bracing model e Base shear optimization with 456 bracing model f Base shear optimization with 789 bracing model

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The undamped fundamental natural circular frequency of the nine-storey building with 123 bracing (\(\omega _{1})\) was estimated to be 6.7869 rad/s. As shown in Fig. 11a, dampers were added, optimally, to the 5th, 6th, 7th and 8th storeys for the top displacement minimization in a nine-storey structure with 123 bracing. Figure 11b shows the damper placement for the base shear minimization in the same model where the dampers were distributed in decreasing quantity from the 4th storey to the upper storeys.

Fig. 11
figure 11

Optimal and uniform damper designs for; a top displacement minimization with 123 bracing model b base shear minimization with 123 bracing model c top displacement minimization with 456 bracing model d base shear minimization with 456 bracing model e top displacement minimization with 789 bracing model f base shear minimization with 789 bracing model

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The undamped fundamental natural circular frequency of the nine-storey building with 456 bracing (\(\omega _{1}\)) was estimated to be 6.4732 rad/s. Dampers were added predominantly to the 2nd, 7th, and 8th storeys for top displacement minimization in the nine-storey structure with 456 bracing as shown in Fig. 11c. Figure 11d shows that for the damper placement for the base shear minimization in the 456 bracing model the dampers were distributed in decreasing quantity from the 1st to 4th storey and the less quantity of damping coefficients are also placed on the 7th, 8th and 9th storeys.

The undamped fundamental natural circular frequency of nine-storey building with 789 bracing (\(\omega _{1})\) was estimated to be 5.5845 rad/s. As shown in Fig. 11e, added dampers for top displacement minimization in nine-storey structure with 456 bracing were focused on 2nd, 3rd, and 4th storeys where the interstorey drifts is large. The damper allocation for the base shear minimization in the 456 bracing model is presented in Fig. 11f showing that the dampers were located in decreasing quantity from the 1st to 6th storey.

5.3 Earthquake response analysis

The seismic response of the 9-storey structure to the Kobe earthquake motion records is shown in Fig. 12. The responses before and after the insertion of the optimal viscous dampers using the ABCA was computed by direct time integration of the model structure.

Fig. 12
figure 12

Kobe earthquake ground motion acceleration record (NS)

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Figure 13a shows the comparison of the optimal damper design based on top storey displacement, the optimal damper design based on the base shear and for the design without dampers. The maximum top storey displacement in both the top displacement optimization and base shear optimization decreased by about 17.33 % and 14.78 % respectively according to whether dampers were applied or not (Fig. 13a). The reduction in the maximum base shear response of the model structure without bracing was approximately 27 % whereas, in the damper design based on top displacement minimization, it decreased by approximately 28 % in case of the damper design based on the base shear, as shown in Fig. 14.

Fig. 13
figure 13

Maximum storey displacement of the 9-storey planar building frame (a) without bracing model (b) 123 bracing model (c) 456 bracing model (d) 789 bracing model

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Fig. 14
figure 14

Maximum base shear response of 9-storey planar building frame

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The maximum displacements on the storey level are shown in Fig. 13b–d for the 123, 456 and 789 bracings. As can be seen there was a similar displacement response from the braced storeys in the three stiff storeys. Furthermore, comparisons of the optimal damper design for the top displacement and base shear in terms of the top displacements in the 123, 456 and 789 bracing models reveal that the optimal damper design for top displacement was more effective than the optimal damper design for the base shear. The maximum bases shear response of the model structures are presented in Fig. 14 this shows that the base shear performance of the added damper models based on the base shear was more effective than the damper models based on the top displacement. Each of the optimal damper design satisfied its own objective. The optimal dampers designs obtained in frequency domain using the ABCA were tested by time history analysis using the Kobe (NS) earthquake acceleration data and the time history analyses showed that the optimal damper designs performed well under the Kobe (NS) earthquake conditions.

6 Conclusions

The ABCA has proved to be a useful design method for the placement of viscous dampers in a planar building frame. In order to discover the optimal damper placement, an optimization problem defined by transfer functions evaluated the fundamental natural circular frequency of the structure was solved by the proposed ABCA without using gradient information. The optimal damper coefficients of the added dampers and the response of the top displacement and base shear of the structure determined by ABCA were compared with the results of the SDSA gradient based algorithm. In addition, after the optimal damper designs based on two objective functions obtained were tested by the time history analyses the ABCA was shown to perform well. This method is very practical and highly effective in determining the optimal placement of the added dampers. It was observed that the use of the ABCA may be a simple way of solving the problem of optimal damper distribution. If an alternative performance function is required to regulate the seismic response of multi-storey shear buildings, the proposed ABCA can be modified for a new function.